Whisk the dashi, egg yolk, miso, sugar, mirin and sake together in a small bowl.

Place sauce in a small saucepan and cook over low heat until it thickens like gravy, stirring constantly.

Heat the oil in a large skillet and cook eggplant pieces, first cooking cut-side down until they brown nicely, then turning them over, adding the sake-water mix, and continuing to cook until they are fork-tender.